Electroweak symmetry breaking in the light of LHC

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(1)Electroweak symmetry breaking in the light of LHC Bogna Kubik. To cite this version: Bogna Kubik. Electroweak symmetry breaking in the light of LHC. Physics [physics]. Université Claude Bernard - Lyon I, 2012. English. �NNT : 2012LYO10136�. �tel-00770109v2�. HAL Id: tel-00770109 https://tel.archives-ouvertes.fr/tel-00770109v2 Submitted on 3 Sep 2015. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) Thése de doctorat presentée devant L’Univérsité Claude Bernard Lyon - I École Doctorale de Physique et d’Astrophysique. Spécialité PHYSIQUE THÉORIQUE / PHYSIQUE DES PARTICULES présentée par Mlle. BOGNA KUBIK en vue de l’obtention du grade de DOCTEUR de L’UNIVERSITÉ CLAUDE BERNARD (Lyon 1). Symétrie électrofaible à la lumière du LHC. Soutenue publiquement le 5 Octobre 2012 devant la commission d’examen formée de : M. Mme. M. Mme. M. M. M.. J. G. Ch. N. A. G. A.. Gascon Bélanger Grojean Mahmoudi Arbey Cacciapaglia Deandrea. Président du jury Rapporteuse Rapporteur Examinateur Examinateur Directeur de thèse Directeur de thèse. -. IPN Lyon LAPTH Annecy CERN Genève LPC Clermont-Ferrand CRAL Lyon IPN Lyon IPN Lyon.

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(4) PhD thesis delivered by Claude Bernard University of Lyon. Speciality: THEORETICAL PHYSICS / PARTICLE PHYSICS submitted by Mrs. BOGNA KUBIK for the degree of DOCTOR OF PHILOSOPHY. Electroweak symmetry breaking in the light of LHC. defended October 5th, 2012 in front of the following Examining Committee: Mr. Mrs. Mr. Mrs. Mr. Mr. Mr.. J. G. Ch. N. A. G. A.. Gascon Bélanger Grojean Mahmoudi Arbey Cacciapaglia Deandrea. President Reviewer Reviewer Examiner Examiner Supervisor Supervisor. -. IPN Lyon LAPTH Annecy CERN Genève LPC Clermont-Ferrand CRAL Lyon IPN Lyon IPN Lyon.

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(6) Aknowledgments This work was realized in the University of Lyon in the Laboratory of Nuclear Physics (IPNL) thanks to the research grant founded by the Ministry of Higher Education and Research. I would like to thanks the two successive directors of IPNL, Bernard Ille and Guy Chanfray for stay in the IPNL I could have. I would like to thank especially to my PhD advisors, Aldo Deandrea and Giacomo Cacciapaglia, for supporting me during these past three years. You have been supportive and have given me the freedom to pursue various projects without objection. You have also provided insightful discussions about the research. I am also very grateful for the scientific advices and knowledge and many insightful discussions and suggestions. The research projects and discussions that we had together allowed me the insertion in the scientific word not only French but in the entire world. Thanks for your enthusiasm and joy during my PhD! I also have to thank Jules Gascon for accepting to be the president of my PhD committee and for many useful remarks about my work. I also thank my two referees, Geneviève Bélanger and Christophe Grojean for the time they dedicated for reading my thesis and also for their very smart remarks, questions and opinions. Finally I have to thank the members of my PhD committee, Nazila and Alexandre for the time to go to Lyon and listening my presentation. I also thank my collaborator, Alexandre A., for his help and ideas about the dark matter stuff. Thanks to Sacha D. for discussions and helping in first steps in Lyon. I would also like to thanks the Antares group in the IFIC in Valencia and especially Juanjo Hernandez-Rey for the possibility of spending some time in the experimental group and visiting Spain. Also, thanks to the program Multidark I could go the meeting in Madrid. Thanks to all my colleagues in Spain for the great time I had there. Finally I have to thank to my colleagues from 334 - Jeremy L., Gregoire G. and Ahmad T. for the imagination, invention, joyfulness, political formation and musical sessions. Thanks to Nico C., Vincent J., Nico B. and Clem B. for introducing me in the real life of the IPNL students. Finally thanks Olivier B., Benoit M. and Antonio U. for the adventure with antiquarks! Thanks to Max and Julien for the "soirées" unforgettable. Thanks to Luca for the MadGraph help and to Antoine for being my "parrain". Special thanks to the IPNL stuff for the help (especially for the answers on my "informatic" questions and problems). Na koniec dzi¸ekuje moim rodzicom za wszystko.. iii.

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(8) Abstract The extra-dimensional extensions of the Standard Model of particles are now in a very active epoch of development. The motivations of introducing extra dimensions are based on one hand on string theories that require the existence of new dimensions to be consistent. On the other hand such theories can potentially explain the hierarchy problem, number of fermion generations, proton stability and other enigmas of the Standard Model. The common feature of these models is that they provide a new neutral weakly interacting particle - perfect candidate to the Dark Matter. It’s stability is preserved by the so-called KK parity which prohibits the decays of the LKP into SM particles. The geometry of the underlying space determines the particle spectrum of the model, thus the mass and the spin of the DM candidate, which in turn plays the key role in the phenomenological studies We present a model with two universal extra dimensions compactified on a real projective plane. This particular geometry is chosen because chiral fermions can be defined on such orbifold and the stability of the neutral dark matter candidate arise naturally from the intrinsic geometrical properties of the space without adding any new symmetries ad hoc. We present the particle spectrum at loop order up to the second level in Kaluza-Klein expansion. The particularity of the spectrum is that the mass splittings within each KK level are highly degenerated providing a very interesting potential signatures in the LHC. We study the dark matter phenomenology in our model and constrain the parameter space by comparing our results with WMAP data and direct detection experiments. Using the obtained bounds we focus on the collider phenomenology of our model.. v.

(9) ABSTRACT. vi.

(10) Contents Introduction. 1. I. 5. Elementary Particles and Interactions - Introduction. 1 Mathematics, Geometry, and 1.1 Mathematics . . . . . . . . 1.2 Geometry . . . . . . . . . . 1.3 Space and Time . . . . . . .. Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Standard Models in (micro and macro) Physics 2.1 The Standard Model of Particle Physics . . . . . . . . . 2.1.1 Gauge Invariance Principle - Gauge Theories . . 2.1.2 SM lagrangian . . . . . . . . . . . . . . . . . . . 2.1.3 Precision tests of the Standard Model predictions 2.1.4 Standard Model is not a final story . . . . . . . . 2.1.5 Possible extensions . . . . . . . . . . . . . . . . . 2.2 Standard model of cosmology . . . . . . . . . . . . . . . 2.2.1 Friedmann-Robertson-Walker model . . . . . . . 2.2.2 Short history of the universe . . . . . . . . . . . 3 Dark Matter - Where? What? Why? 3.1 Evidences for the dark Matter . . . . . . . . . 3.1.1 Galactic scale . . . . . . . . . . . . . . 3.1.2 Galaxy clusters scale . . . . . . . . . . 3.1.3 Cosmological scale . . . . . . . . . . . 3.2 Dark matter candidates . . . . . . . . . . . . 3.3 Detection Schemes . . . . . . . . . . . . . . . 3.3.1 Direct searches . . . . . . . . . . . . . 3.3.2 Indirect searches . . . . . . . . . . . . 3.4 Relic abundance of dark matter . . . . . . . . 3.4.1 The standard case of relic abundance 3.4.2 Case with co-annihilations . . . . . . . vii. . . . . . . . . . . . . . . . . . . . . . . . . . . . only . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . annihilations . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . .. 7 7 8 9. . . . . . . . . .. 11 11 11 13 17 20 22 24 24 25. . . . . . . . . . . .. 27 27 28 29 30 33 36 36 39 41 41 43.

(11) CONTENTS. II. Extra Dimensions - models and techniques. 45. 4 Compactified space general description of the extra dimensional models 4.1 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 One dimensional orbifolds and manifolds . . . . . . . . . . . . . . . . . 4.1.3 Two dimensional orbifolds and manifolds . . . . . . . . . . . . . . . . 4.2 Quantum fields in the extra dimensions . . . . . . . . . . . . . . . . . . . . . 4.2.1 Basics of Kaluza Klein decomposition . . . . . . . . . . . . . . . . . . 4.2.2 Orbifold or Interval? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Fermions on the circle and on the interval - chirality problem . . . . . 4.2.4 Gauge fields on S 1 and S 1 /Z2 . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Couplings of gauge modes . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Radiative corrections in extra dimensions . . . . . . . . . . . . . . . . . . . . 4.3.1 Methods for loop calculation . . . . . . . . . . . . . . . . . . . . . . . 4.4 Extra dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Large extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Warped extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . .. III. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. Study of the 6D Model on the Real Projective Plane. 5 General presentation of the model 5.1 Real projective plane - orbifold construction . . . . . . 5.1.1 Basic definitions . . . . . . . . . . . . . . . . . 5.1.2 Localized terms on the RP 2 and the KK parity 5.2 Quantum Fields on the real projective plane . . . . . . 5.2.1 Scalar field . . . . . . . . . . . . . . . . . . . . 5.2.2 Gauge field . . . . . . . . . . . . . . . . . . . . 5.2.3 Spinorial field . . . . . . . . . . . . . . . . . . . 5.2.4 Higgs boson . . . . . . . . . . . . . . . . . . . . 5.2.5 Yukawa couplings for fermions . . . . . . . . . 5.2.6 Standard Model on the real projective plane . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 47 47 49 49 50 52 52 53 56 56 58 59 59 63 63 64. 67 . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 69 69 69 71 73 73 74 78 83 84 85. 6 Mass spectrum of the (2,0) - (0,2) modes at loop level 6.1 Divergences due to compactification - general remarks . . . . 6.2 Winding modes on the real projective plane . . . . . . . . . . 6.3 One loop level spectrum of the (2, 0) − (0, 2) tiers using the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Loop corrections to the (2, 0) − (0, 2) gauge bosons . . 6.3.2 Loop corrections to the (2, 0) − (0, 2) fermions . . . . 6.3.3 Heavy higgs bosons . . . . . . . . . . . . . . . . . . . . 6.3.4 Full mass spectrum at loop level - summary . . . . . .. . . . . . . . . . . . . . . . . . . . . . . mixed propagator . . . . . . . . . . . 91 . . . . . . . . . . . 92 . . . . . . . . . . . 102 . . . . . . . . . . . 105 . . . . . . . . . . . 108. 7 Relic Abundance of Dark Matter 7.1 Relic abundance - analytical results . . . 7.1.1 Annihilations into gauge bosons . 7.1.2 Annihilations into fermions . . . 7.1.3 Relic abundance – annihilations .. . . . .. viii. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 87 87 88. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 111 113 113 116 118.

(12) CONTENTS. 7.2. 7.3. 7.4 7.5. 7.1.4 Relic abundance – co-annihilation effects . . . . . . . . . . . . Relic abundance - numerical results . . . . . . . . . . . . . . . . . . . 7.2.1 L1 scenario - relic abundance at tree level . . . . . . . . . . . 7.2.2 L1 scenario - relic abundance at loop level . . . . . . . . . . . 7.2.3 L2 scenario - relic abundance . . . . . . . . . . . . . . . . . . 7.2.4 Comparison of the mKK bounds in L1 and L2 scenarios . . . Cut-off dependence of the relic abundance . . . . . . . . . . . . . . . 7.3.1 Cut-off dependence of the relic abundance – tree level . . . . 7.3.2 Cut-off dependence of the relic abundance – loop level . . . . H 2 localized mass parameter mloc dependence of the relic abundance Direct detection bounds . . . . . . . . . . . . . . . . . . . . . . . . .. 8 LHC phenomenology 8.1 Decays of the (2,0) KK modes - influence of the 8.1.1 (2,0) gauge bosons . . . . . . . . . . . . 8.1.2 (2,0) leptons . . . . . . . . . . . . . . . 8.1.3 (2,0) quarks . . . . . . . . . . . . . . . . 8.2 Heavy states production in the LHC . . . . . . 8.3 LHC signatures . . . . . . . . . . . . . . . . . . 8.3.1 W → l± νl searches . . . . . . . . . . . . 8.3.2 W → W Z searches . . . . . . . . . . . 8.3.3 Z → l+ l− searches . . . . . . . . . . . . 8.3.4 Di-jet resonances . . . . . . . . . . . . . 8.4 Summary of all the bounds on the mKK . . . .. geometry of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. the orbifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 121 127 127 128 131 132 134 136 138 139 143. . . . . . . . . . . .. 147 147 147 151 153 156 160 161 162 163 163 164. Conclusion & outlook. 167. A Notations. 169. B Annihilation Cross Sections 171 (1) B.1 Co-annihilations A(1) with leptons lS/D and ν (1) . . . . . . . . . . . . . . . . . . . 171 (1). B.2 Co-annihilations of leptons lS/D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 (1) (1) B.3 Co-annihilations of lepton - antilepton lS/D ¯lS/D . . . . . . . . . . . . . . . . . . . 175. ix.

(13) CONTENTS. x.

(14) List of Figures 2.1 2.2 2.3 2.4. Experimental constraints on top and higgs Electroweak observables global fit . . . . . Constraints on STU parameters . . . . . . Evolution of the universe . . . . . . . . .. . . . .. 18 19 21 26. 3.1 3.2 3.3 3.4 3.5 3.6. Rotation curves of galaxies . . . . . . . . . . . . . . . . Gravitational lensing from Chandra X-ray and Hubble Field Planetary Camera . . . . . . . . . . . . . . . . . . CMB anisotropies . . . . . . . . . . . . . . . . . . . . . . Cosmological parameters fit . . . . . . . . . . . . . . . . Direct detection bounds on WIMP cross section . . . . . dark matter decoupling . . . . . . . . . . . . . . . . . .. . . . . . . Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . Wide . . . . . . . . . . . . . . . . . . . .. 28. 4.1 4.2 4.3 4.4 4.5. Example of pattern of a pgg symmetry group . . . . . . . . . Fundamental domains of the torus and Klein bottle orbifolds Fundamental domains of the RP 2 and Chiral Square orbifolds Spectrum of a scalar field on one-dimensional orbifolds . . . . Spectrum of a fermionic field on an interval . . . . . . . . . .. . . . . .. . . . . .. . . . . .. 50 51 52 54 57. 5.1. Geometry of the RP 2 orbifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70. 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9. Cross level two point functions for gauge bosons . . . . . Electroweak mixing angle . . . . . . . . . . . . . . . . . . KK gauge bosons mass splittings . . . . . . . . . . . . . . Radiative corrections of fermion masses . . . . . . . . . . KK top quarks mixing angle . . . . . . . . . . . . . . . . . KK lepton mass splittings . . . . . . . . . . . . . . . . . . KK quarks mass splittings . . . . . . . . . . . . . . . . . . KK higgs mass splittings . . . . . . . . . . . . . . . . . . . Bound for mloc /mKK values from electroweak constraints. 7.1 7.2 7.3 7.4 7.5 7.6. Annihilations A(1) A(1) → ZZ . . . . . . . . . . . . . . . . . . . . . . . . Annihilations A(1) A(1) → W W . . . . . . . . . . . . . . . . . . . . . . . Annihilations A(1) A(1) → HH . . . . . . . . . . . . . . . . . . . . . . . Annihilations A(1) A(1) → f f¯ . . . . . . . . . . . . . . . . . . . . . . . . Relic Abundance - annihilations - analytical . . . . . . . . . . . . . . . . Non-relativistic expansion of the effective cross section - arel coefficients nihilations into bosons and fermions . . . . . . . . . . . . . . . . . . . . Relative contributions of annihilations into bosons and into fermions . .. 7.7. xi. masses . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . Space . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . .. . . . . . . . . .. . . . .. . . . . . . . . .. . . . .. . . . . .. . . . . . . . . .. . . . .. . . . . .. . . . . . . . . .. . . . .. . . . . .. . . . . . . . . .. . . . .. . . . . .. . . . . . . . . .. . . . .. . . . . .. . . . . . . . . .. . . . .. . . . . . . . . .. . . . . . . . . . . for . . . .. . . . .. . . . . .. . . . . . . . . .. . . . .. . . . . .. . . . . . . . . .. 30 31 33 37 41. . . . . . . . . .. 98 101 102 102 104 105 105 106 107. . . . . . . . . . . . . . . . an. . . . . .. 114 115 116 116 118 119 120.

(15) LIST OF FIGURES. 7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 7.28 7.29. Mass splittings of (1,0) level for R4 R5 . . . . . . . . . . . . . . . . . . . . . . Relic Abundance - co-annihilations e1S - analytical . . . . . . . . . . . . . . . . . 1 Relic abundance - co-annihilations lS/D - analytical . . . . . . . . . . . . . . . . . Relic abundance in mUED model . . . . . . . . . . . . . . . . . . . . . . . . . . . L1 scenario tree level processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . L1 scenario loop level processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relic abundance in L1 scenario - numerical . . . . . . . . . . . . . . . . . . . . . Relative contributions of different annihilation cross sections to the relic abundance in L1 and L2 scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L2 scenario tree level processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . L2 scenario loop level processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relic abundance in L2 scenario - numerical . . . . . . . . . . . . . . . . . . . . . Relative contributions of the partial annihilation cross sections to the relic abundance in L2 scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relic abundance in L1 and L2 scenarios - summary . . . . . . . . . . . . . . . . . Mass splittings of the (1,0) particles as a function of ΛR . . . . . . . . . . . . . . Cutoff dependence of Ωh2 in L1 scenario R4 R5 . . . . . . . . . . . . . . . . . Cutoff dependence of Ωh2 in L2 scenario R4 R5 . . . . . . . . . . . . . . . . . Total annihilation cross section A(1) A(1) → SM as a function of pcms . . . . . . . Bound on mloc as a function of the mKK mass . . . . . . . . . . . . . . . . . . . Relic abundance dependence on the mloc in the L1 scenario . . . . . . . . . . . . Relic abundance dependence on the mloc in the L2 scenario . . . . . . . . . . . . Direct detection - diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bounds on mKK from direct detection experiments . . . . . . . . . . . . . . . . .. 133 134 135 136 137 138 141 141 142 143 144. 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14. Total widths of KK gauge bosons . . . . . . . . . . . . . . . . . . Branching ratios of A(2) and Z (2) into SM for R4 R5 . . . . . Branching ratios of Z (2) into heavy KK states for R4 R5 . . . Branching ratios of W (2) into SM for R4 R5 . . . . . . . . . . Branching ratios of G(2) for R4 R5 . . . . . . . . . . . . . . . . Total widths of KK leptons for R4 R5 . . . . . . . . . . . . . . Branching ratios of KK leptons for R4 R5 . . . . . . . . . . . . Total widths of KK quarks for R4 R5 . . . . . . . . . . . . . . Total production cross section of (2,0) colored states in the LHC Single production cross section of (2,0) gauge boson in the LHC . Bounds from W → lν . . . . . . . . . . . . . . . . . . . . . . . . Bounds from W → W Z signals . . . . . . . . . . . . . . . . . . . Bounds from di-lepton events Z → l= l− . . . . . . . . . . . . . . Bounds from di-jet resonances . . . . . . . . . . . . . . . . . . . .. 148 150 150 151 152 152 153 155 158 158 162 162 163 164. 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 121 123 124 126 127 128 130 130 131 131 133. (1). B.1 Co-annihilations A(1) lS/D into neutral gauge bosons . . . . . . . . . . . . . . . . 171 (1). B.2 Co-annihilations A(1) lS/D into W gauge bosons . . . . . . . . . . . . . . . . . . . 172 (1). B.3 Co-annihilations A(1) lS/D into Higgs boson H . . . . . . . . . . . . . . . . . . . . 172 (1). (1). B.4 Co-annihilations lS/D lS/D into ll . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 (1). B.5 Co-annihilations lD ν (1) into lν . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 (1) (1) B.6 Annihilations lS/D ¯lS/D into l¯l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174. xii.

(16) List of Tables 2.1. Standard Model content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14. 3.1. Direct detection bounds on WIMP cross section . . . . . . . . . . . . . . . . . . .. 38. 4.1 4.2. Definitions of one-dimensional symmetry groups . . . . . . . . . . . . . . . . . . Definitions of two-dimensional symmetry groups . . . . . . . . . . . . . . . . . .. 50 51. 5.1 5.2 5.3 5.4 5.5. Wave functions of a Aμ gauge boson in the (+−) gauge boson in the Wave functions of a Aμ (−+) Wave functions of a Aμ gauge boson in the (−−) Wave functions of a Aμ gauge boson in the Standard Model content on the RP 2 . . . . .. 6.1 6.2 6.3 6.4 6.5 6.6. Radiative corrections to (n, 0) gauge boson masses Numerical values of Φi (n) . . . . . . . . . . . . . . Radiative corrections to (0, n) gauge boson masses Diagonal corrections to fermion masses . . . . . . . Mass spectrum of the (1,0) KK level . . . . . . . . Mass spectrum of the (2,0) KK level . . . . . . . .. (++). unitary gauge unitary gauge unitary gauge unitary gauge . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . 97 . 97 . 97 . 103 . 108 . 109. Couplings of A(1) with fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bounds on mKK from relic abundance 1 . . . . . . . . . . . . . . . . . . . . . . . Co-annihilation cross sections - magnitudes . . . . . . . . . . . . . . . . . . . . . Bounds on mKK from co-annihilations . . . . . . . . . . . . . . . . . . . . . . . . Branching ratios of the (2, 0) KK modes into SM particles . . . . . . . . . . . . . Bounds on the mKK form the relic abundance calculation in the models L1 and L2 - summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Mass splittings of the (1,0) particles as a function of ΛR . . . . . . . . . . . . . . 7.8 Relic abundance bounds for mKK in L1 and L2 scenarios for R4 R5 geometry - summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Relic abundance bounds for mKK in L1 and L2 scenarios for R4 = R5 geometry - summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Branching ratios of H (2) as a function of mloc . . . . . . . . . . . . . . . . . . . . 7.11 Resonant values of mKK as a function of mloc . . . . . . . . . . . . . . . . . . . . 7.12 Bounds on mKK from direct detection experiments . . . . . . . . . . . . . . . . .. 7.1 7.2 7.3 7.4 7.5 7.6. 8.1 8.2. 76 77 77 78 86. 117 118 123 123 132 134 135 138 139 140 140 145. Decay channels and branching ratios of KK gauge bosons for R4 R5 . . . . . . 148 Decay channels and branching ratios of KK gauge bosons for R4 = R5 . . . . . . 149 xiii.

(17) LIST OF TABLES 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11. Branching ratios of KK gauge bosons into SM for R4 R5 . . . Branching ratios of KK gauge bosons into SM for R4 = R5 . . . Decay channels and branching ratios of KK leptons for R4 R5 Decay channels and branching ratios of KK leptons for R4 = R5 Decay channels and branching ratios of KK quarks for R4 R5 Decay channels and branching ratios of KK quarks for R4 = R5 . Total production cross section of (2,0) colored states in the LHC LHC bounds on the mass scale mKK . . . . . . . . . . . . . . . . Bounds on the mKK - summary . . . . . . . . . . . . . . . . . . .. xiv. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 149 149 153 153 156 157 158 165 165.

(18) Introduction My adventure with particle physics begun rather early, when I was in college, with a famous book of Leon M. Lederman "The God Particle“. I red when ill lying in the bed. Since then the world of elementary particles, and especially the ”god particle“ motivated and guided all my scientific career. Now I am really pleased that I can contribute in the community of physicists and put a small brick into a great construction that is particle physics. The history of particle physics is rather recent. It begins about 1890s when new discoveries have caused significant paradigm shifts that pushed scientists to review the basic concepts that were considered as a very solid background of all the scientific theories. The General Relativity of Einstein (1916) revealed some questions about the nature of space and time and the geometry of our Universe. Even more dangerous consequences had the discovery of quantum mechanics. This theory raised in the minds of physicists confronted to the problems such as the Black Body radiation explained by Planck in 1900 or photoelectric effect described by Einstein in 1905. These discoveries were only a prelude before the great epoch of particle physics when efforts of thousands of theorists and experimentalists from “both sides of the scale” were put together with the common goal - to explain the nature of the Universe. “Both sides of the scale” refers to the micro and macro physics because, that is remarkable, some explanations of the very microscopic world can come from the observations and theories at the cosmological scale. Now our best understanding of how the basic constituents of the Universe behave, is encapsulated in the Standard Model of Particles. Formulated in the early 1970s it has successfully explained a host of experimental results and precisely predicted a wide variety of phenomena. Over time and through many experiments by many physicists, the Standard Model has become established as a well-tested physics theory. Its power has been confirmed once more with the observation a peek that can be interpreted as a Higgs boson - the key to the origin of particle mass. Although the discovery of Higgs does not write the final ending to the story. Even though the Standard Model is currently the best description we have of the subatomic world, it does not explain the complete picture. The theory incorporates only three out of the four fundamental forces, omitting gravity. There are also important questions it cannot answer, such as what is dark matter, what happened to the missing antimatter, and more. In order to study the limits of this theory, new experiments in high energy physics and in observational cosmology have been developed in the last decade. The results from Large Hadron Collider (LHC), a proton-proton collider located at CERN, are now the most expected to reveal what is hidden above the Standard Model domain of predictability. For sure the experimental data using the high energies reached by the LHC can push knowledge forward, challenging those who seek confirmation of established knowledge, and those who dare to dream beyond the paradigm. 1.

(19) INTRODUCTION. As high energies are required, the Universe itself can accelerate our investigation as it is home to numerous exotic and beautiful phenomena, some of which can generate almost inconceivable amounts of energy. Supermassive black holes, merging neutron stars, streams of hot gas moving close to the speed of light are only few examples of phenomena that generate gamma-ray radiation, the most energetic form of radiation. What is the origin of such high energies? Studying these energetic objects add to our understanding of the nature of the Universe and how it behaves. These are the goals of FERMI telescope. At the cosmological scale WMAP mission was proposed to NASA in 1995. It was launched in 2001 and is still collecting data from the cosmic microwave background radiation - the radiant heat left over from the Big Bang. The properties of the radiation contain a wealth of information about physical conditions in the early universe and a great deal of effort has gone into measuring those properties since its discovery. The role of theorists in all this world of experiments is to guide them. To analyze data in the LHC one has to know where to search a weak signal hidden in a huge QCD background. Therefore a huge effort is made by theorists to make predictions as precise as possible and to suggest what could be the nature of new physics potentially visible at LHC. A part of this thesis will aim to give some new directions that could be explored in the collider. On the other hand we use the cosmological parameters derived from WMAP observations to put some bounds on our model. I started the three-year period of doctorate with one principal goal - to understand the microscopic nature of the Universe, to investigate the problems that arise within the newest theories and the motivations that push thousands of physicists to not stop searching. I had an opportunity to familiarize myself with the supersymmetry formalism during my Master studies. I must say that the mathematical construction of superfields and superspace seems very attractive to me and particularly beautiful in its simplicity. Then I started studying extra-dimensional models that propose a completely different mathematical formulation of the new physics and thus the nature of new predicted constituents would be somehow distinct from those predicted by supersymmetry. In the first year of my PhD I joined a group that was already working on a model based on a six-dimensional space-time with two flat extra dimensions compactified on a real projective plane. The main attractive feature of this model that motivates our work is that the geometrical properties of the underlying space provide a stable dark matter candidate in a natural way. The existence of extra dimensions will manifest itself in our four-dimensional space as a tower of Kaluza-Klein states propagating as “ordinary” particles. It is however not sufficient to stop here. The main goal of our team was to study the phenomenology of the model in both, the LHC collider environment and in the dark matter sector. To this aim we need to proceed through some steps that are summarized in this thesis. The document divided in three parts: – Part I describes the basic models of physics that are currently used and that are a framework of our study. In this part there are three chapters. – In the fist chapter we introduce some notions about the mathematical language used in physics and its motivations. – In Chapter 2 we present the Standard Models of Particles and of Cosmology that put 2.

(20) bounds on our models on one hand and on the other had have some problems that our model tries to solve. – In Chapter 3 is dedicated to Dark Matter phenomenology. Here we summarize the current experimental searches of Dark Matter particle and we present the relic abundance calculation method that we used in our study. – Part II introduces tools for working with extra-dimensional models. We present the construction of the simplest orbifolds and quantum fields propagating in extra dimensions. We introduce the notion of Kaluza Klein states also. Then we present the methods used in loop calculation in extra dimensions. The main classes of extra-dimensional models known in the literature are presented at the end of this part. – Part III presents the model and our results. This part is divided in three chapters. – Chapter 5 is completely dedicated to the description of the model. We present the construction of the orbifold first. Then we define the quantum fields and give the tree level spectrum of particles present in the scenario. – Chapter 6 presents the calculation of the radiative corrections to the (2,0)-(0,2) KaluzaKlein states. Using the techniques presented in the Part II we have calculated the spectrum of the model at one loop level for two different geometries of the orbifold R4 R5 and R4 = R5 . The main feature of the spectrum of the model is underlined - a very small mass splittings within each KK level. This will have interesting consequences in the phenomenological signatures of our model. We introduce also the free parameter mloc - a localized mass of Higgs bosons that will allow in the later study to change considerably the bounds of the compactification scale. – Chapter 7 focuses on the Dark Matter phenomenology. We study the bounds on the compactification scale coming from the relic abundance WMAP data. First we show in analytical manner the impact of co-annihilations on the relic abundance bounds. They are expected to be strong as the mass splittings negligible at least for KK leptons. In the next step we perform a full numerical study of the model using MicrOMEGAs. We include all the loop induced couplings and use loop-level spectrum of the model to predict the precise bounds on the compactification scale in both geometries R4 R5 and R4 = R5 . Then we try to vary parameters of the model to investigate their impact on the relic abundance. Finally we present the bounds coming from the direct detection experiments. – Chapter 8 is dedicated to the LHC phenomenology. We identify the main interesting channels that are studied in the LHC experiments and that could potentially give signatures of our model in collider searches. We infer bounds on the compactification scale from those processes in two considered geometries R4 R5 and R4 = R5 . To finish this chapter we summarize all the numerical bounds on the compactification scale we have obtained in our analysis. – In the Appendices we give our notation conventions and summarize the analytical results of the annihilation cross sections used in the main chapters.. 3.

(21) INTRODUCTION. 4.

(22) Part I. Elementary Particles and Interactions - Introduction. 5.

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(24) Chapter 1. Mathematics, Geometry, and Space-Time 1.1. Mathematics. By convention there is color, By convention sweetness, By convention bitterness, But in reality there are atoms and space" -Democritus (circa 400 BCE). Mathematics is a natural science that describes the most general forms of processes in nature as assumes Galileo. Mathematics deals with reality, postulating axioms believed to be true and confirmed pragmatically by the empirical truth of the theorems derived from them. It distinguishes itself from other natural sciences only in that it obtains very few concepts and relations directly from experience, and infers from them the laws of more complex phenomena by purely deductive means. The idea of securing knowledge by logical deduction from unquestionable principles was explicitly proposed by Aristotle, and successfully applied by Euclid in his Elements and later on by Galileo, Newton and their successors. From its inception in ancient Greece, and again in modern times, science adopted a mathematical interpretation of nature. This problem of mathematical interpretation is much deeper than could be seen at first sight and many philosophers, mathematicians and physicists, now also neuropsychologists join the team that discuss whether the Universe has an intrinsic mathematical nature or it can only be accurately described by mathematical formulas used as a language. Einstein wondered how it was possible that mathematics, a product of human thought, so admirably described reality. Mathematical science is “unreasonably effective” in describing physical reality and it predicts with an extreme accuracy the natural processes. In his 1933 Oxford lecture, Einstein highlighted the physical, empirical nature of mathematics by inviting us to consider Euclidean geometry as the science of the possible mutual relations between practically rigid bodies in space – in other words, to treat geometry as a physical science, without abstracting from its original empirical content. Without considering geometry as a natural science he could have never formulated the theory of relativity. Supporting the view of mathematics as a natural science that describes the logic of nature, mathematics often pre-discovers physical reality. For example Dirac’s equation predicted the positron. Moreover, 7.

(25) CHAPTER 1. MATHEMATICS, GEOMETRY, AND SPACE-TIME the fact that natural forms can be described by abstract mathematical concepts suggested to Pythagoras and Heraclitus, and later to Galileo and modern scientists, that mathematics describes the "logic" of the universe.. 1.2. Geometry. "La géométrie euclidienne s’accorde assez bien avec les propriétés des solides naturels, ces corps dont se rapprochent nos membres et notre œil et avec lesquels nous faisons nos instruments de mesure." H.Poincaré “La valeur de la science”. I think that it was the discovery of non-Euclidean geometries in mathematics that has led to the theory that space is not Euclidean or Galilean. The Euclidean axiom of parallels, which was the basis of the axiom system of Euclid, was causing serious concern to mathematicians. This axiom states that for any plane on which there is a line L and a point P that does not lies on the line, there is in the same plane a unique line L which passes through P and is parallel to L which means that two lines in a plain can have at most one common point. In the XIX-th century the mathematicians tried to derive this statement from other axioms of the Euclidean system. It turned out that the statement about two lines on a plane was independent from others axioms and as such it could not be derived as a theorem but must be admitted as another axiom. It was then straight forward to postulate another axioms and derive other geometrical systems. We can cite as examples: 1. Riemannian geometry: for any plane on which there is a line L and a point P that does not lie on the line there is no line L parallel to L that passes through P . 2. Lobachevsky geometry: for any plane on which there is a line L and a point P that does not lie on the line there are more than one line L parallel to L that pass through P (and then it can be demonstrated that if there is more than one L parallel to L then there must be an infinite number of lines parallel to L passing through P .) This was this purely axiomatic view of geometries (that even has a name of “pure geometry“) that was objected by Einstein in his conference "Geometry and experience” where he postulated that we should abandon the difference between the “pure geometry” and ”applied geometry“ as the main reason for which we started to study geometry was the need or desire to describe the physical phenomena. On cosmic scales, the only force expected to be relevant is gravity. The first theory of gravitatys, derived by Newton, was embedded later by Einstein into the General Relativity (GR). However, GR is relevant only for describing gravitational forces between bodies which have relative motions comparable to the speed of light. In most other cases, Newton’s gravity gives a sufficiently accurate description. The speed of neighboring galaxies is always much smaller than the speed of light. So, a priori, Newtonian gravity should be able to explain the Hubble flow. One could even think that historically, Newton’s law led to the prediction of the Universe expansion, or at least, to its first interpretation. Amazingly, and for reasons which are more mathematical than physical, it happened not to be the case: the first attempts to describe the global dynamics of the Universe came with GR, in the 1910’s. Newton himself did the first step in the argumentation. He noticed that if the Universe was of finite size, and governed by the law of gravity, then all massive bodies would unavoidably concentrate into a 8.

(26) 1.3. SPACE AND TIME single point, just because of gravitational attraction. If instead it was infinite, and with an approximately homogeneous distribution at initial time, it could concentrate into several points, like planets and stars, because there would be no center to fall in. In that case, the motion of each massive body would be driven by the sum of an infinite number of gravitational forces. Since the mathematics of that time did not allow to deal with this situation, Newton did not proceed with his argument (97). When Einstein tried to build a theory of gravitation compatible with the invariance of the speed of light, he found that the minimal price to pay was to abandon the idea of a gravitational potential, related to the distribution of matter, and whose gradient gives the gravitational field in any point, to assume that our four-dimensional space-time is curved by the presence of matter, to impose that free-falling objects describe geodesics in this space-time (97).. 1.3. Space and Time. "L’experience nous a appris qu’il est plus commode d’attribuer trois dimensions à l’espace." H.Poincaré “La valeur de la science”. The problem of the number of dimensions of our space-time is not new. We can cite some of the great philosophers and scientists in the history who tried to answer this question: 1. Johannes Kepler (1571 – 1630, German mathematician, astronomer and astrologer) In "Mysterium Cosmographicum" (1595): As he indicated in the title, Kepler thought he had revealed God’s geometrical plan for the universe. Much of Kepler’s enthusiasm for the Copernican system stemmed from his theological convictions about the connection between the physical and the spiritual; the universe itself was an image of God, with the Sun corresponding to the Father, the stellar sphere to the Son, and the intervening space between to the Holy Spirit. His first manuscript of Mysterium contained an extensive chapter reconciling heliocentrism with biblical passages that seemed to support geocentrism. 2. Gottfried Leibniz (1646 – 1716, German philosopher-mathematician) Rather than being an entity that independently exists over and above other matter, Leibniz held that space is no more than the collection of spatial relations between objects in the world: "space is that which results from places taken together". Unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an idealized abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people 3. Sir Isaac Newton (1642 – 1727 English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian) For Newton space and time are absolute. 4. Immanuel Kant (1724 – 1804, German philosopher) In the eighteenth century the German philosopher Immanuel Kant developed a theory where he refuses to see the space and time as objective features of the world but as a framework that helps us to organize our experiences: "Space and time are the framework 9.

(27) CHAPTER 1. MATHEMATICS, GEOMETRY, AND SPACE-TIME within which the mind is constrained to construct its experience of reality." ("Critique of Pure Reason -chpt. Transcendental Aesthetic"). The idea that our space-time could have more than three dimensions dates back to the 1920s. Then Theodor Kaluza (1919) and Oscar Klein (1926) published their works on the unification of gravity and electromagnetism in five dimensions - the only two forces known at that time. The discovery of new forces and particles during the following years revealed the Kaluza-Klein theory not sufficient and the theory was forgotten for a while. The revival of extra-dimensional theories was due to the birth of (super)string theories in the 1970s and 1980s. However the extra dimensions introduced on the grounds of superstring theory are expected to be very small, they have a scale of about MP−1 ∼ 10−35 m, and thus there was no hope for probing such small scales in existing and upcoming experiments. The new ideas begun to emerge in the 1990s. In 1990 Antoniadis explains the supersymmetry breaking by introducing a large extra dimension at a scale of TeV−1 m. In the late 1990s the ADD scenario, the warped space model and some inventions in the string theory (D-branes) and M-theory propose the existence of new dimensions at the scales accessible in current searches at colliders. In this thesis we will adopt the modest point of view and assume that mathematics is a powerful language which can describe the natural phenomena. We will present the mathematical construction of the fundamental models of the Universe that aim to embed in simple mathematical formulas the processes that we observe in micro and macro scale. Then, we will be less conservative, from the "classical physics" point of view, and we will assume the existence of two additional dimensions. The mathematical language will allow to describe this assumption in terms of physical observables - a Kaluza Klein states. We will then study what are the implications of such assumption.. 10.

(28) Chapter 2. Standard Models in (micro and macro) Physics 2.1. The Standard Model of Particle Physics. The Standard Model of particle physics (SM) describes the interactions of quarks and leptons that are the constituents of all matter we know about. The strong interactions are described by quantum chromodynamics (QCD) while the electromagnetic and the weak interactions are described by the electroweak theory. This theory has proven to be very successful in describing a tremendous variety of experimental data ranging over many decades of energy. The discovery of neutral currents in the 1970s followed by the direct observation of the W and Z bosons at the CERN Sp¯ pS collider in the early 1980s confirmed the ideas underlying the electroweak framework. Since then, precision measurements of the properties of the W and Z bosons at both e+ e− and hadron colliders have allowed a test of electroweak theory at the 10−3 level. QCD has been tested in the perturbative regime in hard collision processes that result in the breakup of the colliding hadrons. The main and most mysterious feature of the SM is that particles are associated to the mathematical beings, a state vectors Ψ with given transformation properties under the Lorentz group. This tight relation between a physical particle as it is observed indirectly in the accelerator detectors and the intrinsic properties of a mathematical operator associated to the particle is very profound. The group structure of the SM is even more exciting, as for a long time physicists looked suspiciously on the particle ZOO that was discovered in the first accelerator searches.. 2.1.1. Gauge Invariance Principle - Gauge Theories. The dynamic arises from a symmetry principle. If we require the lagrangian to be invariant under local gauge transformations we are forced to introduce a set of gauge fields with couplings to elementary scalar and fermion matter that are completely determined by symmetry properties (up to global symmetry transformations). Below we describe the construction of a general Young-Mills theory which is the basis of the construction of the Standard Model.. Consider the lagrangian density L [φ, ∂μ φ] which is invariant under a D-dimensional continuous group of transformations Γ: 11.

(29) CHAPTER 2. STANDARD MODELS IN (MICRO AND MACRO) PHYSICS. φ = U (θA )φ. with A = 1, 2, . . . , D. (2.1). . For θA infinitesimal, U (θA ) 1 + ig A θA T A , where T A are the generators of Γ in the representation of the fields φ. We restrict ourselves to the case of internal symmetries, so that the matrices T A are independent of the space-time coordinates. The generators T A are normalized in such a way that for the lowest dimensional non-trivial representation {tA } of Γ we have 1 tr[tA tB ] = δ AB 2. (2.2). The generators satisfy the commutation relations: . . T A , T B = iCABC T C. (2.3). where CABC are the structure constants. If the transformation parameters θA depend on the space-time coordinates, θA = θA (xμ ), then the lagrangian density L [φ, ∂μ φ] is, in general, A no longer invariant under the gauge transformations U θ (xμ ) and should be replaced by L [φ, Dμ φ] where the ordinary derivative is replaced by a covariant derivative Dμ Dμ = ∂μ + igVμ. (2.4). . where we define Vμ = A T A VμA for a set of D gauge fields {VμA } (in a one-to-one correspondence with the group generators) with the transformation law Vμ. = U Vμ U. −1. . +. i (∂μ U )U −1 g. (2.5). for constant θA , V reduces to a tensor of the adjoint representation of Γ Vμ = U Vμ U −1 Vμ + ig[θ, Vμ ]. (2.6). which implies that . VμA = VμA − gCABC θB VμC. (2.7). Equations 2.4 and 2.5 ensure that φ and Dμ φ have the same transformation properties (Dμ φ) = U (Dμ φ). (2.8). The gauge-invariant kinetic term for the gauge fields Vμ is constructed in terms of the field strength 12.

(30) 2.1. THE STANDARD MODEL OF PARTICLE PHYSICS. A Fμν = ∂μ VνA − ∂ν VμA − gCABC VμB VνC. (2.9). that transforms as a tensor of the adjoint representation = U Fμν U −1 Fμν. (2.10). The complete Yang-Mills lagrangian invariant under gauge transformations can be written in the form: LY M =. 1 A A μν F F + L[φ, Dμ φ] 4 A μν. (2.11). For an abelian theory the gauge transformation reduces to U [θ(xμ )] = exp[ieQθ(xμ )], where Q is the charge generator. The associated gauge field, according to equation 2.5, transforms as Vμ = Vμ − ∂μ θ(x). (2.12). In this case the field strength Fμν is linear in Vμ , so that in the absence of matter the theory A tensor contains both linear and is free. On the other hand, in the non-abelian case, the Fμν A quadratic terms in Vμ , so the theory is non-trivial even in the absence of matter.. 2.1.2. SM lagrangian. The SM is a non-abelian gauge theory based on the gauge group SU(3)C × SU(2)L × U(1)Y . The SU(3)C is the Quantum Chromodynamics (QCD) gauge group, which governs the strong interactions. SU(2)L × U(1)Y is the gauge group that unifies the weak and electromagnetic forces and is spontaneously broken to U(1)em by the higgs mechanism. The SU(3)C is assumed to be unbroken. The matter fields of the Standard Model are the three generations of quarks {(u, d), (c, s), (t, b)} and leptons {(e,νe ),(μ,νμ ),(τ ,ντ )} as well as three kinds of gauge bosons {Bμ , Wμa , GA μ } that mediate the three interactions described by the Standard Model. There is also a scalar particle, the higgs boson Φ, that is responsible for the electroweak symmetry breaking. In order to allow for a chiral structure for the weak interactions, the left- and right-handed components of quark and lepton fields are assigned to different representations of the electroweak gauge group SU(2)L × U(1)Y . Thus, the mass terms for fermions, of the form ψ¯L ψR +h.c. are forbidden in the symmetric limit (here ψL and ψR refer to a doublet and singlet of SU(2)L respectively, that belong to the same generation). The SU(3)C gauge bosons are the gluons and the resulting gauge theory is the Quantum Chromodynamics (QCD). Quarks are assigned to the fundamental 3 representation. Thus antiquarks are assigned to the conjugate ¯ 3 representation. All other particles are SU(3)C singlets, and do not directly couple to gluons. The SU(3)C × SU(2)L × U(1)Y assignments for the matter fields of the first generation of quarks and leptons are shown in the table 2.1. Other generations are copies of this in that they have the same quantum numbers. 13.

(31) CHAPTER 2. STANDARD MODELS IN (MICRO AND MACRO) PHYSICS Field. L=. νL eL. uL dL. SU (2)L. U (1)Y. 1. 2. -1. 1. 1. -2. U (1)em. . . eR. . SU (3)C. . 0 −1. . +1. . 2 3 − 13. 3. 2. 1 3. uR. 3. 1. 4 3. 2 3. dR. 3. 1. − 23. - 13. Φ. 1. 2. 1. 0. Bμ. 1. 1. 0. 0. Wμa. 1. 3. 0. 0, ±1. GA μ. 8. 1. 0. 0. Q=. . Table 2.1: Dimensions of representations and charges of fermions of the first generation. Fermions of the second and the third generations have the same quantum numbers.. The lagrangian of the Standard Model is given by: LSM = Lgauge + Lmatter + Lhiggs + LY ukawa. (2.13). Gauge sector The kinetic terms of the gauge fields have the compact form 3 8 1 1 1 a Wμν W a μν − GA GA μν Lgauge = − Bμν B μν − 4 4 a=1 4 A=1 μν. (2.14). a and GA are the field strengths associated to the U(1) , SU(2) and SU(3) where Bμν Wμν Y L C μν respectively. Bμν. = ∂μ Bν − ∂ν Bμ. a Wμν. = ∂μ Wνa − ∂ν Wμa − g2 abc Wμb Wνc. GA μν. C = ∂μ Gaν − ∂ν Gaμ − gs f ABC GB μ Gν. (2.15). The g2 and g3 are the coupling constants of SU(2)L and SU(3)C gauge groups respectively. The SU (2)L structure constants abc form the totally antisymmetric Levi-Civita tensor with 123 = +1. The f abc are the SU(3)C structure constants.. 14.

(32) 2.1. THE STANDARD MODEL OF PARTICLE PHYSICS Matter sector The matter lagrangian is given by Lmatter =. . . iL† σ ¯μ Dμ L + ie†R σμ Dμ eR + iQ† σ ¯μ Dμ Q + iu†R σμ Dμ uR + id†R σμ Dμ dR. . generations. (2.16) where the covariant derivatives have the form 3 8 y λA A A a a tL/R Wμ − ig3 θ3 Gμ Dμ = ∂μ − ig1 Bμ − ig2 2 2 a=1 A=1. (2.17). where YL/R /2 and tA L/R are the SU (2)L and U (1)Y generators, respectively, in the reducible representations ψL/R . λA are the generators of the SU(3)C algebra, which in the fundamental representation correspond to the Gell-Mann matrices. θ3 = +1, 0 for triplets and singlets of SU(3)C respectively.. Higgs sector and electroweak symmetry breaking The electroweak symmetry breaking sector of the SM is particularly simple, and consists of a single complex SU(2)L doublet Φ of spin zero fields with gauge quantum numbers shown in the table 2.1. Φ must be a doublet in order to write down the gauge invariant mass terms for fermions in the Yukawa sector. The field Φ acquires a VEV (Vacuum Expectation Value) signaling the spontaneous breakdown of the electroweak symmetry. This VEV is left invariant by one combination of SU(2)L and U(1)Y generators which generates a different U(1) group which is identified as U(1)em . The corresponding linear combination of gauge fields remains massless and is identified with the photon Aμ . The dynamics of the higgs field Φ is governed by the lagrangian Lhiggs = |Dμ Φ|2 − V (Φ). (2.18). where the covariant derivative of the higgs field has the form 3 y taL Wμa Dμ = ∂μ − ig1 Bμ − ig2 2 a=1. (2.19). The potential V (Φ) is the most general renormalizable, gauge invariant polynomial of degree 4 with two parameters μ2 , λ > 0 V (Φ) = −μ2 Φ† Φ + λ(Φ† Φ)2. (2.20). The Φ fields receives the vacuum expectation value at the minimum of the potential V (Φ) at 2 |Φ|2 = μ2λ = v 2 . One then expands the higgs field around the minimal value . Φ=. φ+ 1 √ (v + h + iφ0 ) 2 15. (2.21).

(33) CHAPTER 2. STANDARD MODELS IN (MICRO AND MACRO) PHYSICS Introducing the above expansion of Φ into the lagrangian LSM one finds the physical states, Aμ , a pair of charged massive spin 1 bosons W ± and a massive spin 1 neutral boson Z 0 . Their masses that can be written in terms of the initial gauge fields as Aμ. = sin θW Wμ3 + cos θW Bμ. Zμ. = − cos θW Wμ3 + sin θW Bμ. Wμ±. 1 √ 2. =.

(34). with sin θW = g2 / formulas. Wμ1. g12 + g22 and cos θW = g1 /. ∓.

(35). iWμ2. (2.22). g12 + g22 . The masses then are given by the. 1 1

(36) 2 mW = g2 v g1 + g22 v (2.23) 2 2 The neutral scalar higgs boson h, which is left over as the relic of spontaneously broken symmetry, receives a mass m2h = 2λv 2 . mA = 0. mZ =. Yukawa sector and the CKM matrix The fermion masses are generated in the Yukawa sector by interactions of fermionic fields with the higgs boson Φ. LY ukawa = −. . Y u Q† (iσ2 Φ∗)uR −. generations. . Y d Q† ΦdR −. generations. . Y e L† ΦeR + h.c. (2.24). generations. where Y u,d,e are the three 3 × 3 complex Yukawa matrices. In order to find the mass eigenstates, as for the gauge fields, one introduces the higgs field expansion defined in the equation 2.21 into the above formula. Then to diagonalize LY ukawa it is necessary to introduce unitary matrices related to the Yukawa matrices by a unitary transformation u Ydiag = VLu Y u VRu†. d Ydiag = VLd Y d VRd†. e Ydiag = VLe Y e VRe†. (2.25). The physical states are then given by uL = VLu† uL. dL = VLd† dL. uR = VRu† uR. dR = VRd† dR. eL = VLe† eL. νL = VLe† νL. (2.26). eR = VRe† eR u,d,e . and their diagonal mass matrices are M u,d,e = √v2 Ydiag What results from this diagonalization are the mixings between different generations of quarks. Looking at the quark part of Lmatter one can easily see that the charged currents are now modified as. g2 g2 Lmatter ⊃ − √ u†L σ ¯ μ Wμ+ dL + h.c = − √ uL† VLu VLd† σ ¯ μ Wμ+ dL + h.c 2 2 16. (2.27).

(37) 2.1. THE STANDARD MODEL OF PARTICLE PHYSICS where the Cabibbo-Kobayashi-Maskawa matrix VCKM = VLu VLd† is a unitary matrix that mixes the three quark generations. The CKM matrix definition contains only one physical phase which is the source of CP-violation in the quark sector of the Standard Model.. 2.1.3. Precision tests of the Standard Model predictions and parameters. The success of the Standard Model lies in the extreme accuracy of its predictions of physical phenomena form the atomic scales down to scales of about 10−18 m. The predictions of the SM have been probed by Tevatron and LEP up the the scales of order of few hundred GeV. The impressive predictability of the SM is a consequence of the renormalizabitliy of its parameters. There are 19 parameters within the SM that remain unexplained and that are chosen to fit the data. In the electroweak gauge sector they include the three gauge coupling constants g1 , g2 and g3 for the three gauge groups U(1)Y , SU(2)L and SU(3)C respectively (equivalently three g2 g12 e2 with e = g2 sin θW and sin2 θW = g2 +g other parameters can be used: αs = 4π3 , αew = 4π 2 ). 1 2 In the matter sector there are nine parameters associated with the masses of charged fermions and four mixing angles in the CKM matrix. Moreover there are two parameters associated with higgs boson: the higgs VEV v and the quartic coupling λ. The remaining one is the θQCD parameter. Moreover, we should also care about the parameters in the neutrino sector, as the data from neutrino oscillation experiments provide convincing evidence for neutrino masses. With 3 light Majorana neutrinos there are at least 9 additional parameters in the neutrino sector: 3 masses and 6 mixing angles and phases. Those parameters have to be constrained with an outstanding precision by various experiments that probe the SM observables and the influence of new physics should be investigated in order to constrain the possible extensions of the SM as we shall see later. As the SM is a model that has been constructed in not so far past and its mathematical construction seems very efficient and relatively simple, many experiments have been dedicated to investigate the predictions of this exciting theory. The main progress in the domain was made in LEP e+ e− collider in the 1990’s and has already given first bounds on the top and the higgs masses. The observations of the top quark was later confirmed by Tevatron data in 1995. Recently new results from ATLAS and CMS collaborations released new data providing an indication of a higgs boson with the mass of about 125 GeV as illustrated in the figure 2.1. At LEP 1 and SLC, there were high-precision measurements of various Z pole observables (8), (55). These include the Z mass and total width ΓZ and partial widths into fermions Γf f¯. In the table 2.2 the referenced values are given also for the partial width into hadrons Γhad , charged leptons Γl+ l− and the width for invisible decays Γinv . The latter can be used to determine the number of neutrino flavors much lighter than the m2Z to be Nν = Γinv /Γth (ν ν¯) = 2.984 ± 0.009 for (mt , mH ) = (173.4, 117). The other observables refer to σhad ≡ 12πΓe+ e− Γhad /m2Z Γ2Z and the branching ratios Rf ≡ Γf f¯/Γhad for hadrons and analogous expressions for Rl related to leptons. The Rl parameters are especially useful to constrain the αs value. The measurements of the left-right asymmetry ALR ≡. σL − σR σL + σR. (2.28). has been measures by SLD and SLC (55) collaborations and can heavily constrain the sin θW parameter. The observables that were measured indirectly include the Af parameters, where 17.

(38) CHAPTER 2. STANDARD MODELS IN (MICRO AND MACRO) PHYSICS Mass of the Top Quark. 14. 0. July 2010. (* preliminary). CDF-I dilepton. 167.4 ±11.4 (±10.3 ± 4.9). DØ-I dilepton. 168.4 ±12.8 (±12.3 ± 3.6). CDF-II dilepton *. 170.6 ± 3.8. (± 2.2 ± 3.1). DØ-II dilepton *. 174.7 ± 3.8. (± 2.9 ± 2.4). CDF-I lepton+jets. 176.1 ± 7.4. (± 5.1 ± 5.3). DØ-I lepton+jets. 180.1 ± 5.3. (± 3.9 ± 3.6). CDF-II lepton+jets *. 173.0 ± 1.2. (± 0.7 ± 1.1). DØ-II lepton+jets *. 173.7 ± 1.8. (± 0.8 ± 1.6). CDF-I alljets. 186.0 ±11.5 (±10.0 ± 5.7). CDF-II alljets. 174.8 ± 2.5. (± 1.7 ± 1.9). CDF-II track. 175.3 ± 6.9. (± 6.2 ± 3.0). Tevatron combination *. 173.3 ± 1.1. (± 0.6 ± 0.9). CDF March’07. χ2/dof 6.1/10 (81%) 12.4 ± =2.7 (± 1.5 ± 2.2). 150. 160. 170 180 mtop (GeV/c2). 190. ( ± stat ± syst). 200. Figure 2.1: Left panel. Combined results D0 and CFD on the top quark mass. Extracted from (78). Right. panel:The observed (full line) and expected (dashed line) 95% CL combined upper limits on the SM higgs boson production cross section divided by the Standard Model expectation as a function of mh in the low mass range of this analysis. The dashed curves show the median expected limit in the absence of a signal and the green and yellow bands indicate the corresponding 68% and 95%. Extracted from (5).. f = e, μ, τ, b, c, s defined as Af ≡. f 2¯ gVf g¯A. f2 g¯Vf 2 + g¯A. (2.29). (0,f ). and the forward-backward asymmetries AF B = 43 Ae Af .. f parameters in the above formulas refer to vector and axial-vector couplings The g¯Vf and g¯A in the SM lagrangian after the electroweak symmetry braking and are defined as. gVf. ≡ tf3L − 2qf sin2 θW. f gA ≡ tf3L. (2.30) (2.31). and here tf3L is the weak isospin of a given fermion and qf is its charge in units of e. In the f correspond to the effective values where the electroweak radiative has bar quantities g¯Vf and g¯A been taken into account. Finally the s¯2l parameter were extracted from the measurements of forward-backward asymp collisions in D0 and CDF (80) experiments. metries AF B for e+ e− final states in p¯ The global fit results to the experimental data are summarized in the table 2.2 for the main Z-pole observables. The magnitude of the CKM matrix elements has been measured with an extreme precision in the experiments such as BELLE (study of meson mixings) (129) or BABAR (study of B-mesons) (96). Using the set of Wolfenstein parameters (λ, A, ρ, η) the CKM matrix can be written as 18.

(39) 2.1. THE STANDARD MODEL OF PARTICLE PHYSICS. Figure 2.2: The principal Z-pole observables and their SM predictions.The column denoted Pull gives the standard deviations for the principal fit with mH free, while the column denoted Dev. (Deviation) is for mH = 124.5 GeV fixed. Extracted from (72). ⎛. VCKM. ⎞. 1 − λ2 /2 λ Aλ3 (ρ − iη) ⎜ ⎟ −λ 1 − λ2 /2 Aλ2 =⎝ ⎠ Aλ3 (1 − ρ − iη) −Aλ2 1. (2.32). The λ = sin θCabibbo parameter is a critical ingredient in determinations of the other parameters and in tests of CKM unitarity. Current experiments of kaon and hyperon decays suggest a value λ ≈ 0.220 and λ ≈ 0.225 and this discrepancy is discussed in therm of violation of the unitarity condition 2 2 2 | + |V12 | + |V13 |=1 |V11. (2.33). For a more complete review of experiments and constraints on the SM observables one can refer 19.

(40) CHAPTER 2. STANDARD MODELS IN (MICRO AND MACRO) PHYSICS to (72).. 2.1.4. Standard Model is not a final story. The standard electroweak model is a mathematically-consistent renormalizable field theory which predicts many of the experimental facts. It successfully predicted the existence and form of the weak neutral current, the existence and masses of the W and Z bosons, and the charm quark, as necessitated by the GIM mechanism. The charged current weak interactions, as described by the generalized Fermi theory, were successfully incorporated, as was quantum electrodynamics. The consistency between theory and experiment indirectly tested the radiative corrections and ideas of renormalization and allowed the successful prediction of the top quark mass. When combined with quantum chromodynamics for the strong interactions, the Standard Model is almost certainly the approximately correct description of the elementary particles and their interactions down to at least 10−16 cm. When combined with general relativity for classical gravity the SM accounts for most of the observed features of nature. However, the theory has far too much arbitrariness to be the final theory. The new physics models can be tested by the ρ0 parameter ρ0 =. m2W m2Z cos2 θW ρ. (2.34). which describes the new sources of electroweak symmetry breaking that cannot arise within the 2 is the “ default” parameter assuming validity of the SM. Another SM. Here ρ = m2Z /m2W cos θW set of parameters, S, T and U can be used to constrain the many new physics models as well 1 . The T parameter is proportional to the difference between the W and Z self-energies at q 2 = 0 (measures the electroweak symmetry breaking), while S (S + U ) is associated with the difference between the Z (W ) self-energy at q 2 = m2Z . The data allow for a simultaneous determination 2 (from the Z pole asymmetries), S (from m ), U (from m ), T (mainly from Γ ), α of sinθW s Z W Z (from Rl , σhad , and the τ lifetime ττ ), and mt (from the hadron colliders). Assuming 115.5 GeV < mh < 127 GeV the values are (72): S = 0.00+0.11 −0.10 T = 0.02+0.11 −0.12 U = 0.08 ± 0.11. 2 sin θW = 0.23125 ± 0.00016 αs (mZ ) = 0.1197 ± 0.0018 = 173.4 ± 1.0GeV mt. (2.35). These values depend only weakly on the mh mass but can give quite stringent constraints on the exotic extensions of the SM. For example the S parameter can be used to constrain the number of fermion families, under the assumption that there are no new contributions to T or U and therefore that any new families are degenerate. Then an extra generation of SM fermions is excluded at the 5.7 σ level. This restriction can be relaxed by allowing T to vary as well, since T > 0 is expected from a non-degenerate extra family. Then, a fourth family is disfavored but not excluded by the current electroweak precision data. One important consequence of a heavy fourth family is to increase the higgs production cross section by gluon fusion by a factor ∼ 9, which considerably strengthens the exclusion limits from direct searches at the Tevatron (41) and LHC (4). In contrast, heavy degenerate non-chiral fermions (also known as vector-like or 1. There is no simple parametrization to describe the effects of every type of new physics on every possible observable. The S, T, and U formalism describes many types of heavy physics which affect only the gauge self-energies, and it can be applied to all precision observables.. 20.

(41) 2.1. THE STANDARD MODEL OF PARTICLE PHYSICS multiplets), which are predicted in many grand unified theories (72) and other extensions of the SM, do not contribute to S, T , and U (or to ρ0 ), and do not require large coupling constants.. Figure 2.3: 1 σ constraints (39.35%) on S and T parameters from various inputs combined with mZ . S and T represent the contributions of new physics only. The contours assume 115.5 GeV < mh < 127 GeV except for the larger (violet) one for all data which is for 600 GeV < mh < 1 TeV. Extracted from (72). Here we briefly summarize the theoretical enigmas and possible explanations of the Standard Model of fundamental interactions and basic constituents of our world. Gauge group: The Standard Model is a complicated direct product of three subgroups, SU(3)C ×SU(2)L ×U(1)Y , with three independent gauge couplings. There is no explanation for why only the electroweak part is chiral (parity-violating). The charge quantization is left unexplained as well. The complicated gauge structure suggests the existence of some underlying unification of the interactions, such as one would expect in a superstring or GUT theories. Fermions: We know from the every-day life that under ordinary terrestrial conditions all matter can be constructed out of fermions of the first family (u, d, e, νe ). Yet, three families of fermions exist with no obvious role in nature. The number of families nor the huge extent between the masses of fermions, which varies over 5 orders of magnitude between the top quark and the electron, can not be explained within the Standard Model. Even more mysterious are the neutrinos, which are many orders of magnitude lighter. A related difficulty is that while the CP violation observed in the laboratory is well accounted for by the phase in the CKM matrix, there is no SM source of CP breaking adequate to explain the baryon asymmetry of the universe. Naturalness: Another unexplained mass hierarchy exists in the higgs sector. Quantum corrections to the higgs mass give quadratically divergent contributions . . . 1 1 ∼ (2.36) (3g 2 + g12 ) + 6λ − 6yt2 Λ2 + O(log Λ) 2 32π 4 2 Therefore if the Standard Model were valid up to the Planck scale MP = 1.22 × 1019 GeV the “bare“ mass in the higgs lagrangian should be fine tuned with a precision of 10−32 to cancel the radiative corrections and get the higgs mass of order of 100 GeV. δm2h. 21.